Related papers: Some overdetermined problems related to the anisot…

We investigate the $W^{1,p}$ estimates of the Neumann problem for the Schr\"odinger equation $-\Delta u+ V u={\rm div}(f)$ in the region above a convex graph. For any $p>2$, we obtain a sufficient condition for the $W^{1,p}$ solvability. As…

We study the asymptotic behavior, as $\gamma$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$ -\Delta u=\frac{f(x)}{u^\gamma}\,\text{ in }\Omega,…

We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 +…

Let $n \ge 2$, $p \in (1, \, +\infty)$ be given and let $\Sigma$ be a $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$. Denote by $A$ its second fundamental form, and by $\mathring{A}$ the tensor $A - \frac{1}{n} A^i_i g$ where $g…

In this note, as a particular case of a more general result, we obtain the following theorem: Let $\Omega\subseteq {\bf R}^n$ be a non-empty bounded open set and let $f:\overline {\Omega}\to {\bf R}^n$ be a continuous function which is…

The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -\Delta_{p}u & = & \lambda\omega_{1}(x)\left\vert u\right\vert ^{q-2}% u+\beta\omega_{2}(x)\left\vert u\right\vert…

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$.…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where…

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem ${\rm inf}{{\rm ess sup}_{x \in \Omega} f(\nabla u(x)): u \in u_0 + W^{1,\infty}_0(\Omega)}$ when the supremand $f$ is not…

A relatively polynomially convex subset $V$ of a domain $\Omega$ has the extension property if for every polynomial $p$ there is a bounded holomorphic function $\phi$ on $\Omega$ that agrees with $p$ on $V$ and whose $H^\infty$ norm on…

In this work we study the existence of solutions $u \in W^{1,p}_0(\Omega)$ to the implicit elliptic problem $ f(x, u, \nabla u, \Delta_p u)= 0$ in $ \Omega $, where $ \Omega $ is a bounded domain in $ \mathbb R^N $, $ N \ge 2 $, with smooth…

We prove a priori estimates for solutions of order $2$ linear elliptic PDEs in divergence form on subanalytic domains. More precisely, we study the solutions of a strongly elliptic equation $Lu=f$, with $f\in L^2(\mathcal{\Omega})$ and…

We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure $\mu$ with a positive smooth density $f$, any solution to the $L_p$ Minkowski problem in $\mathbb{R}^{n+1}$ with $p \le…

Using Harnack's inequality and a scaling argument we study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $\zeta \in \partial\Omega\cup\{\infty\}$ for the quasilinear elliptic…

We discuss the solvability of Dirichlet problems of the type $- \Delta_{p, w} u = \sigma$ in $\Omega$; $u = 0$ on $\partial \Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $\Delta_{p, w}$ is a weighted $(p, w)$-Laplacian…

For any Wulff shape $\mathcal{W}$, its dual Wulff shape and spherical Wulff shape $\widetilde{\mathcal{W}}$ can be defined naturally. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, we show…

For any convex set $\Omega \subset {\mathbb R} ^N$, we provide a lower bound for the inverse of the Poincar\'e constant in $W ^ {1, 1}(\Omega)$: it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an…

In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-\Delta_p u =…

Let $\alpha\in(0,1)$, $\Omega$ be a bounded open domain in $R^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$ and $\omega$ be the Hausdorff measure on $\partial\Omega$. We denote by $\frac{\partial^\alpha \omega}{\partial…